as a distribution (field) of horizontal subspaces – an Ehresmann connection – and via a connection 11-form which annihilates the distribution of horizontal subspaces. The connection in that sense induces a smooth version of Hurewicz connection.

The usual textbook convention is to say just connection for the distribution of horizontal subspaces, and the objects of the other three approaches one calls more specifically covariant derivative, connection 11-form and parallel transport.

In the remainder of this Idea-section we discuss a bit more how to understand connections in terms of parallel transport.

from the path groupoid of XX to the Atiyah Lie groupoid of PP that is smooth in a suitable sense and splits the Atiyah sequence in that P1(X)→traAt″(X)→P1(X)\mathbf{P}_1(X) \stackrel{tra}{\to} At''(X) \to \mathbf{P}_1(X) (see the notation at Atiyah Lie groupoid).

Terminology

The functor tratra is called the parallel transport of the connection. This terminology comes from looking at the orbits of all points in PP under tratra (i.e. from looking at the category of elements of tratra): these trace out paths in PP sitting over paths in XX and one thinks of the image of a point p∈Pxp \in P_x under tra(γ)tra(\gamma) as the result of propagating pp parallel to these curves in PP.

Flat connections

It may happen that the assignment tra:γ↦tra(γ)tra : \gamma \mapsto tra(\gamma) only depends on the homotopy class of the path γ\gamma relative to its endpoints x,yx, y. In other words: that tratra factors through the functor P1(X)→Π1(X)P_1(X) \to \Pi_1(X) from the path groupoid to the fundamental groupoid of XX. In that case the connection is called a flat connection.

of vector bundles. Locally on XX – meaning: when everything is pulled back to a coverY→XY \to X of XX – this is a Lie(G)Lie(G)-valued 1-form A∈Ω1(Y,Lie(G))A \in \Omega^1(Y, Lie(G)) with certain special properties.

In particular, since every GG-principal bundle canonically trivializes when pulled back to its own total space PP, a connection in this case gives rise to a 1-form A∈Ω1(P)A \in \Omega^1(P) satisfying two conditions. This data is called an Ehresmann connection.

If instead P=EP = E is a vector bundle, then the two conditions satisfies by AA imply that it defines a linear map

Proof

Choose a partition of unity(ρi∈C∞(X,ℝ))(\rho_i \in C^\infty(X,\mathbb{R})) subordinate to the good open cover{Ui→X}\{U_i \to X\} with respect to which a given cocycle g:X→BGg : X \to \mathbf{B}G is expressed: